Foundations of Vietoris homology theory with applications to non-compact spaces
type: CreativeWork
description: One of the two principal objectives of the present work is to provide an underlying structure for Vietoris homology theory as it is used by topologists today. It is hoped that the existence of this structure will give formal status to the theory and enable results to be formulated and proved with a degree of precision not previously attainable. The second objective is to extend the classical theorems of Vietoris homology theory to a broader class of metric spaces by removing the restriction of compactness and replacing it by some weaker requirement. For this purpose it is necessary that the structure provided for the theory be designed in such a way that compactness is not assumed in defining the basic concepts or in establishing relationships among them. Although Vietoris' primary interest was in the mapping theorem, his methods attracted much attention. Lefschetz discussed the Vietoris theory in his 1930 Colloquium Publication [18], relating it to a homology theory of his own and to an early one of Alexandroff [l]. The Vietoris theory was soon applied to other problems by such mathematicians as Whyburn [29], [30] and Eilenberg [10]. Perhaps the most important application was that of Alexandroff who in 1932 published his celebrated paper "Dimensionstheorie. Ein Beftrag zur Geometrie der abgeschlossenen Mengen" [2]. In this paper Alexandroff constructed a homoloical theory of dimension for compact metric spaces and related the homological dimension to the covering dimension of such spaces. He was then able to characterize the covering dimension by means of a homological condition (see Chapter VI of the present work). In the process of developing the homological dimension theory, Alexandroff also developed the homology theory. His paper became and has remained to the present time the standard reference for Vietoris homology theory. Since the publication of the Alexandroff paper there have been only a few discussions of the theory itself. In his 1942 Colloquium Public…
abstract: One of the two principal objectives of the present work is to provide an underlying structure for Vietoris homology theory as it is used by topologists today. It is hoped that the existence of this structure will give formal status to the theory and enable results to be formulated and proved with a degree of precision not previously attainable. The second objective is to extend the classical theorems of Vietoris homology theory to a broader class of metric spaces by removing the restriction of compactness and replacing it by some weaker requirement. For this purpose it is necessary that the structure provided for the theory be designed in such a way that compactness is not assumed in defining the basic concepts or in establishing relationships among them. Although Vietoris' primary interest was in the mapping theorem, his methods attracted much attention. Lefschetz discussed the Vietoris theory in his 1930 Colloquium Publication [18], relating it to a homology theory of his own and to an early one of Alexandroff [l]. The Vietoris theory was soon applied to other problems by such mathematicians as Whyburn [29], [30] and Eilenberg [10]. Perhaps the most important application was that of Alexandroff who in 1932 published his celebrated paper "Dimensionstheorie. Ein Beftrag zur Geometrie der abgeschlossenen Mengen" [2]. In this paper Alexandroff constructed a homoloical theory of dimension for compact metric spaces and related the homological dimension to the covering dimension of such spaces. He was then able to characterize the covering dimension by means of a homological condition (see Chapter VI of the present work). In the process of developing the homological dimension theory, Alexandroff also developed the homology theory. His paper became and has remained to the present time the standard reference for Vietoris homology theory. Since the publication of the Alexandroff paper there have been only a few discussions of the theory itself. In his 1942 Colloquium Public…
author: Reed, Robert Edward
keywords: Homology theory, Algebraic topology, Metric spaces
date published: 1972-01-01T08:00:00Z
date modified: 2026-07-24T01:10:35Z
provider: Binghamton University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Binghamton University